Latinized Roman is a redundancy. And the Greeks were in Southern Italy before the rise of Rome or the Latins.
If Romans were Greek (the inhabitants of Latium were not Romans until after the citizneship law of AD 212), and Greeks of Hellas became Romans through conquest, then the Greeks of Magna Grecia were Romans, and that they now speak a Latin tongue means they were Latinized.
But the most learned of Roman historians, among who is Porcius Cato, who compiled with the greatest care the genealogies of the Italian cities, Gaius Semporonis and many others, say they are Greeks, part of those who once dwelt in Achaia, and migrated many generations before the Trojan war." (Dionysius of Halicarnassus, Roman Antiquities, I, XI)
Certainly your kith and kin in Greece and Turkey and Syria call themselves Romans, or Romaoi, as did and do their Muslim opressors, who fancied themselves as Sultans of Rum, or Rome if you will.